Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence
Social Recommendation Using Low-Rank Semidefinite Program
1
Jianke Zhu1 , Hao Ma2 , Chun Chen1 and Jiajun Bu1
Zhejiang Provincial Key Laboratory of Service Robot, College of Computer Science, Zhejiang University
2
Microsoft Research
{jkzhu, chenc, bjj}@zju.edu.cn, haoma@microsoft.com
important clues of the social connections or the trust relations among the users, which may lead to the undesired
prediction performance for the recommender system with a
very large number of users.
Recently, there are a surge of research interests on the social recommendation (Ma et al. 2008; Ma, King, and Lyu
2009; Jamali and Ester 2010), which tend to employ the user
connections to improve the prediction quality. These methods are dependent on the probabilistic matrix factorization
with the gradient descent optimization. However, the shared
low-dimensional latent subspace in (Ma et al. 2008) cannot
effectively capture the underlying social connections and the
tastes among the users. Although the ensemble method (Ma,
King, and Lyu 2009) alleviates this problem by directly fusing the users’ favors and their trusted friends’ decisions, it
involves with the intensive computation on calculating the
fused prediction value. As the total number of user-item ratings is usually quite large, the training process may become
time-consuming in case of the users with many social connections. Additionally, a weighting parameter has to be set
empirically in order to balance the tastes of the users’ and
their friends’.
Since the existing approaches have the limitations, there
is a need for developing the new techniques to resolve
these challenges. In this paper, we address the current issues
by introducing a low-rank semidefinite program (LRSDP)
approach. LRSDP (Burer and Monteiro 2003; Burer and
Choi 2006) is developed to efficiently solve the rank constrained semidefinite program (SDP), which has the advantages of the low computational complexity and memory requirement. Therefore, it is capable of tackling the critical
challenge of social recommendation with the large scale
sparse data and the extra constraints. In contrast to the previous social recommendation approaches, we directly regularize the user-specific latent space through the graph Laplacian (Cvetkovic, Doob, and Sachs 1998), which is able to
capture the underlying social relationships between the different users. Motivated by the recent work (Mitra, Sheorey,
and Chellappa 2010) in the computer vision domain, we formulate the social recommendation problem into a low-rank
factorization problem with the graph Laplacian regularization, which can be very efficiently solved by an LRSDP
optimization algorithm (Burer and Monteiro 2003). The experimental evaluations on the Epinions dataset show the en-
Abstract
The most critical challenge for the recommendation system is
to achieve the high prediction quality on the large scale sparse
data contributed by the users. In this paper, we present a novel
approach to the social recommendation problem, which takes
the advantage of the graph Laplacian regularization to capture the underlying social relationship among the users. Differently from the previous approaches, that are based on the
conventional gradient descent optimization, we formulate the
presented graph Laplacian regularized social recommendation problem into a low-rank semidefinite program, which is
able to be efficiently solved by the quasi-Newton algorithm.
We have conducted the empirical evaluation on a large scale
dataset of high sparsity, the promising experimental results
show that our method is very effective and efficient for the
social recommendation task.
1
Introduction
As the rapid growth of Web 2.0 sites and Web applications,
the recommender systems have become more and more important for the end users to filter out the large amount of
useless information. Currently, the recommendation is not
only of the great research interest to be explored, but also of
the huge commercial opportunities in practice. During the
past decade, the extensive research efforts in information
retrieval (Linden, Smith, and York 2003; Hofmann 2003),
machine learning (Breese, Heckerman, and Kadie 1998;
Zhang and Koren 2007; Si and Jin 2003) and data mining
domains have devoted to the recommendation related techniques. More importantly, the recommender systems have
already been deployed in the production recommendation at
Amazon and the music recommendation at iTunes.
The recommender systems are mainly based on the technique of collaborative filtering (Breese, Heckerman, and
Kadie 1998), which automatically predicts the interest of an
active user using the information collected from the other
users. In general, the users only rate a very small portion
of the whole item set. Therefore, the grand challenge for
the recommender system is how to deal with the large scale
dataset with lots of missing entries. Moreover, the conventional recommender systems are solely dependent on the information of user-item ratings. Obviously, they ignore the
c 2011, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
158
couraging results of the proposed method comparing to the
state-of-the-art approaches in terms of both prediction accuracy and time complexity.
The rest of this paper is organized as follows. Section 2
goes somewhat deeper into the previous methods for the
recommender system. Section 3 presents the formulation of
recommendation using low-rank matrix factorization. Section 4 proposes our novel approach to tackling the social
recommendation with the graph Laplacian, and presents the
LRSDP optimization scheme. Section 5 describes the details
of our experimental results. Section 6 sets out our conclusion
and addresses some future work.
2
physical meanings, the sharing approach does not reveal the
underlying relations among the users. In (Ma, King, and
Lyu 2009), an ensemble probabilistic matrix factorization
method is presented to carefully take consideration of the
tastes among the users, which demonstrates the state-of-art
performance on the large scale dataset with a large amount
of missing entries. However, directly fusing the social information into the user-item matrix increases the computational
cost during the optimization, especially for the dataset with
a large number of user-trust connections.
Unlike the existing approaches, our presented method
leverages the matrix factorization with a low-rank semidefinite program, which is based on the quasi-Newton method
having the low computational complexity. The key of our
method is to impose the graph Laplacian regularization on
the user-specific latent space to find the optimal latent factor
matrices with the social constraints, which is further formulated into the LRSDP problem without incurring the heavy
computational cost.
Related Work
The recommender systems have already received quite a bit
of attention, which mainly employ the techniques on collaborative filtering. Moreover, the conventional collaborative filtering approaches are either based on the neighborhood assumption or dependent on a learning model. For
the neighborhood-based methods (Linden, Smith, and York
2003; Breese, Heckerman, and Kadie 1998; Jin, Chai, and
Si 2004), they mainly try to search for the similar active users or items in order to predict the ratings. On the
other hand, the model-based approaches (Hofmann 2003;
Zhang and Koren 2007; Si and Jin 2003) aim to take advantage of the statistical and machine learning techniques to
train a compact model from the rating data. Additionally, the
recommendation is viewed as a matrix complete problem,
which can be formulated as a semidefinite program (Srebro,
Rennie, and Jaakkola 2005).
Our work is closely related to the low-rank matrix factorization method for the recommender system, which makes
the assumption that only a very small number of latent factors affect the users’ preferences in the user-item rating matrix. Moreover, a user’s preference vector could be represented by a linear combination of these latent factors. Assuming that the entries in user-item matrix are corrupted by
Gaussian noise, the low-rank matrix could be approximated
by Singular Value Decomposition (SVD). Although the
SVD approaches enjoy the merits of the closed-form solution, it is not very effective to deal with the large-scale rating
matrix having a large amount of missing entries. In addition,
the probabilistic matrix factorization method (Salakhutdinov
and Mnih 2008) is proposed to take advantages of the latent semantic model and the iterative gradient descent optimization. Most recently, the matrix factorization with missing data problem is formulated into a low-rank semidefinite
program, which demonstrates the promising results on the
computer vision task such as structure from motion (Mitra,
Sheorey, and Chellappa 2010).
Note that all the above methods are based on the assumption that the users are independent, and ignores the social
connections among the different users. To solve this issue,
several remedies are proposed to incorporate the social information into the recommender systems. (Ma et al. 2008)
developed a factor analysis method using the probabilistic
graphical model which combines the user-item matrix with
users’ social trust networks by sharing a common latent lowdimensional user-feature subspace. Due to the lack of the
3
Recommendation by Matrix Factorization
Currently, a very effective approach to recommendation
problem is to predict the missing data entries through factorizing the user-item rating matrix into the user-specific
and item-specific matrices (Salakhutdinov and Mnih 2008;
Ma et al. 2008; Ma, King, and Lyu 2009). The premise behind a low-dimensional factor model is that there is only a
small number of latent factors influencing the preferences,
and that a user’s preference vector is dependent on how each
factor applies to that user. Therefore, we aim to predict the
users’ rating value using the product of these recovered latent factors.
Given m users and n items in a recommender system, the
objective of low-rank matrix factorization is to factorize the
user rating matrix M ∈ Rm×n into the d-rank user-specific
latent factor matrix U ∈ Rm×d and item-specific latent factor matrix V ∈ Rn×d respectively. As there are usually a
large number of items in the recommender system, and a
user may only rate a small portion of the whole item set
practically. As a result, the user rating matrix M is generally
very sparse. It is important to note that the rank d is a quite
small number comparing to the total number of the users m
and the items n. To take into account of the large number
of missing elements in M , we usually solve the following
constrained minimization problem (Salakhutdinov and Mnih
2008; Ma et al. 2008):
(1)
min I ⊗ M − U V 2F + λ U 2F + V 2F
U,V
where · 2F denotes the Frobenius norm, and ⊗ represents
the Hadamard element-wise product. I is an indicator matrix
with ones for the user-rated items and zeros for the missing
data, and U 2F and V 2F are the regularization terms in
order to avoid the overfitting issue. The regularization coefficient λ is employed to balance the weight between the data
penalty term and the regularization term, which is usually
set empirically.
Generally speaking, most of the recommender systems
employ integer values from one to rmax to represent the
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where φ(ui , uj ) represents the kernel function with respect
to the two connected users. Moreover, there are three different kernel functions: namely, binary kernel, distance kernel
and heat kernel.
For the binary kernel, φ(ui , uj ) is simply set to one, which
means that each friend contributes equally in the recommendation process. In the case of distance kernel, we try to find a
similarity function which measures the difference of the two
user ui and uj ’ taste. To this end, we can take advantages of
the observations in the user rating matrix M , and calculate
the distance between the two corresponding vectors.
Since the cosine similarity measure cos yields better results in the empirical evaluation, it is chosen to measure
the users’ taste. In this paper, cos is defined as: cos =
u
i uj
ui ·uj ,where ui and uj are the two rows in M . Due to
the large number of missing elements in M , it is meaningful
to measure the users’ taste by considering the ratings of the
same items. Therefore, we only calculate the cosine distance
on the subset of items which both users have rated, which is
equivalent to the vector space similarity defined in (Breese,
Heckerman, and Kadie 1998). Furthermore, the heat kernel
− 1−cos
can be built by φ(ui , uj ) = e 2ς 2 , where ς denotes the
width for a heat kernel, which is set to one empirically.
As suggested in (Li and Yeung 2009), the graph Laplacian is able to be integrated into the matrix factorization by
adding an extra regularization term tr(U LU ) into the minimization problem in Eqn. 3:
min I ⊗ M − ρ U V 2F + λ U 2F + V 2F
user’s ratings on the items. Without loss of generality, we
first transform the ratings in the matrix M into the interval
(0, 1) using the mapping function g(x) as follows:
g(x) =
x − r̄
rmax
(2)
where r̄ denotes the mean of [1, . . . , rmax ]. Then, the logistic function ρ(x) is employed to bound each element of
matrix multiplication U V within the range (0, 1): ρ(x) =
1
1+exp(−x) . We abuse the notation by denoting the elementwise logistic function of a matrix as ρ(A), and rewrite the
minimization problem in Eqn. 1 into the following form:
min I ⊗ M − ρ U V 2F + λ U 2F + V 2F (3)
U,V
where M denotes the users’ rating value transformed by the
mapping function g(x) in Eqn. 2.
The optimization problem in Eqn. 3 aims to minimize the
sum of reconstruction errors with the two quadratic regularization terms, which is usually solved by the gradient descent method (Salakhutdinov and Mnih 2008) or the alternative projection algorithm (Li and Yeung 2009).
4
Social Recommendation by LRSDP
In this section, we present the proposed low-rank semidefinite program approach to social recommendation task. We
first give our formulation of social recommendation with
the graph Laplacian regularization, and then solve the minimization problem by the low-rank semidefinite program optimization.
4.1
U,V
+ηtr(U LU )
Low-Rank Matrix Factorization with Social
Regularization
(4)
where η is the regularization coefficient for the Laplacian
regularization, which can effectively penalize the deviations
of latent user models from each other if they are linked in
the social network.
Let tr(·) denote the trace of a matrix, we reformulate
the regularization term U 2F + V 2F into the trace form
tr(U U ) + tr(V V ). Thus, the above equation can be further written as follows:
min I ⊗ M − ρ U V 2F + λtr(V V )
U,V
(5)
+tr U (λI + ηL)U
Before making a decision on purchasing a new product, we
are used to obtain the valuable recommendations by consulting our friends with the fruitful experiences. To incorporate
the social relationship information into the recommender
system, we consider to penalize the differences between the
user-specific latent feature vectors when there exist the direct connections between them.
There are several choices to impose the penalty on the
user-specific feature space. In this paper, we propose to directly regularize the user-specific latent space through the
graph Laplacian (Cvetkovic, Doob, and Sachs 1998), which
has the solid theoretical foundation with many successful
applications on manifold learning. Furthermore, the graph
Laplacian can be viewed as a kind of discrete Laplace operator, which is capable of capturing the underlying relationships between the similar users.
Specifically, a graph Laplacian is defined as
It can be clearly observed that the social information is naturally encoded into the regularization for the user-specific
latent factors. More importantly, the above formulation for
the graph Laplacian regularized low-rank matrix factorization is very similar to the optimization problem defined in
Section 3.
L = diag(S1) − S
4.2
where 1 denotes a vector with all one elements. Moreover,
S ∈ Rm×m is a similarity matrix. Let ui and uj denote
the two users having the social connections, and then each
element Si,j is calculated by:
φ(ui , uj ), ui and uj are friends,
Sij = Sji =
0,
otherwise,
LRSDP Optimization
Differently from the previous methods, that are based on
either gradient descent optimization or the alternative projection algorithm, we propose a low-rank semidefinite programming (LRSDP) approach to the matrix factorization
with graph Laplacian regularization. LRSDP (Burer and
Monteiro 2003; Burer and Choi 2006) is essentially a rank
160
Since all the rating values are mapped into the interval (0, 1),
we can utilize the inverse of logistic function and rewrite the
equality constraints in Eqn. 8 as follows:
constrained semidefinite programing (SDP), which is based
on a quasi-Newton algorithm having the lower computational complexity and memory requirement than that of
Newton’s method. It is worthy of mentioning that the graph
Laplacian constraints presented in Section 4.1 can be easily
integrated into the LRSDP optimization framework.
As in (Boyd and Vandenberghe 2004), a standard SDP
optimization has a set of linear equality constraints, and a
matrix non-negativity constraint on an n × n matrix variable
X, which is defined as follows:
min tr C X
s. t. tr A
i = 1, . . . , p
(6)
i X = bi ,
X0
tr A
= log
k GG
tr C X =
Finally, we can write the low-rank matrix factorization
with the graph Laplacian regularization into the following
LRSDP problem:
min tr C GG
Ci,j Xij
G
In (Mitra, Sheorey, and Chellappa 2010), a relaxation
approach is proposed to solve the low-rank matrix factorization problem by changing the variables X = GG .
G ∈ R(m+n)×d is a real matrix which stacks the variables
U and V as follows:
U
G=
V
GG =
UU
V U
UV VV
(9)
Then, we define an (m + n) × (m + n) symmetric matrix C
as below:
0
λI(m×m) + ηL
C=
0
I(n×n)
i,j=1
Therefore,
k = 1, . . . , p
To integrate the graph Laplacian regularization into the
LRSDP optimization framework, we take advantage of the
property for matrix trace tr(ABC) = tr(BCA). Thus, the
user-specific regularization term in Equ. 3 is reformulated
into the following form:
tr U (λI + ηL)U = tr (λI + ηL)U U where C and A1 , . . . , Ap are n × n real symmetric matrices, and b is a p dimensional vector. Note that the objective
function is the form of a general real-valued linear function
in the n × n symmetric matrix space:
n
Mij
,
1 − Mij
s. t.
tr A
= log
k GG
Mij
,
1 − Mij
k = 1, . . . , p
(10)
In this paper, the minimization problem in Eqn. 10 is
solved by the augmented Lagrangian method proposed
in (Burer and Monteiro 2003; Burer and Choi 2006), which
is a nonlinear method using BFGS optimization.
5
Experimental Results
In this section, we present the details of our experimental
implementation and report the empirical results on the social
recommendation task.
As discussed in (Mitra, Sheorey, and Chellappa 2010),
the regularization
term U 2F + V 2F is equivalent to
tr GG . Consequently, the minimization of matrix factorization in Eqn. 1 can be formulated into the following
LRSDP problem:
min tr GG
G
= Mi,j
, k = 1, . . . , p (7)
s. t. tr A
k GG
5.1
Experimental Setup
We evaluate the performance of our presented LRSDP with
graph Laplacian regularization approach on the Epinions
dataset and compare it against several algorithms, such as
Probabilistic Matrix Factorization (PMF), SoRec (Ma et al.
2008), recommendation purely by the trusted friends (Trust)
and Social Trust Ensemble (STE) in (Ma, King, and Lyu
2009). Note that the alternative projection algorithm (Li and
Yeung 2009) was designed for factorizing the dense matrix,
which cannot be directly applied to the social recommendation problem with large scale sparse data.
In this paper, the Epinions dataset is employed as the
testbed for the empirical evaluation on recommendation
with social relation information. Epinions.com is a wellknown review website which is established for sharing the
knowledge on products. At Epinions, the visitors can utilize the reviews on a variety of items in order to assist hem
on making the purchase decisions. They first register an account for free, and then begin submitting reviews according to their own personal opinions that may earn some reward and recognition. To post a review, users need to rate
where Ak are sparse matrices with the non-zero entries at
indices (i, j + m) and (j + m, i) equal to 0.5, and p is the
total number of users’ ratings in M .
In contrast to the original matrix factorization problem defined in Eqn. 1, we need take the extra considerations of the
logistic function and the graph Laplacian regularization for
the proposed social recommendation approach. In the following, we present the remedies to solve these issues.
As for the matrix factorization with the logistic function
ρ(x), we find that the minimization problem in Eqn. 3 is
equivalent to:
min tr GG
G
= Mi,j
, k = 1, . . . , p (8)
s. t. tr A
k ρ GG
161
like the previous methods, we can estimate d based on our
LRSDP formulation. For the proposed approach, we observe
that the number of variables is d(m + n) and the total number of equality constraints is equal to the number of ratings
p. To guarantee a well-posed problem, the rank d should be
p
. Moreover, we did not take into account of
less than m+n
both the inactive users and those items having no ratings in
practice. Therefore, the rank d is set to five.
In this paper, we employ the user-trust information from
Epinions dataset to build the graph Laplacian matrix L. As
discussed in Section 4.1, we have three choices on selecting
the kernel function for the graph Laplacian. Table 3 presents
the experimental results of different settings. From the empirical results, we first observe that both the distance kernel
and heat kernel outperform the binary kernel, which indicates that it is effective to incorporate the users’ taste by
comparing their ratings. Second, the distance kernel function
using the modified cosine similarity achieves the best performance in the empirical evaluation. Therefore, we choose the
modified cosine distance function to build the graph Laplacian matrix L in the following experiments.
Table 1: Statistics of User-Item Rating Matrix of Epinions
Statistics
User Item
Max. Num. of Ratings 1960 7082
Avg. Num. of Ratings 12.21 7.56
Table 2: Statistics of Social Trust Network of Epinions
Statistics
Trust per User Be Trusted per User
Max. Num.
1763
2443
Avg. Num.
9.91
9.91
the product or service with an integer rating value from one
to five. Each member of Epinions maintains a trust list which
presents a network of trust relationships among users.
The Epinions dataset consists of 51,670 users who have
rated a total of 83,509 items. The total number of ratings is
631,064, and the total number of the issued trust statements
is 511,799. Note that the user-item rating matrix of Epinions
dataset is quite sparse, and its density is around 0.015%. The
statistics of user-item matrix and user-trust matrix are summarized into Table 1 and Table 2;
The different amounts of training data (80% and 90%)
are used to evaluate the algorithms. The training data 80%
means that we randomly choose 80% of ratings from the
Epinions dataset as the training data to predict the ratings
of the remaining 20%. The random selection was performed
five times independently. For the proposed method, the regularization coefficient λ is empirically set to one, and η is
0.1. All of our experiments were carried on a PC with Intel
2.8GHz processor and 4GB RAM.
5.2
Table 3: Evaluation on the different kernel functions (A
Smaller RMSE or MAE Value Means a Better Performance)
Training Data Metrics Binary Distance
Heat
RMSE 1.1379
1.1304
1.1323
80%
MAE
0.8639
0.8557
0.8606
RMSE 1.1155
1.1095
1.1197
90%
MAE
0.8540
0.8338
0.8457
Then, we conduct the empirical comparisons to investigate the effectiveness of the logistic function ρ(x) for normalization. Moreover, we study the LRSDP method without the graph Laplacian regularization, which is equivalent
to the low-rank semidefinite program solution for the minimization problem in Eqn. 3. To make it clear, the proposed LRSDP with the graph Laplacian regularization approach is denoted as “LRSDP”. The LRSDP method without normalization is denoted as “LRSDP(w/o ρ)”. Similarly,
the LRSDP method without the graph Laplacian regularization is denoted as “LRSDP(w/o L)”. Table 4 shows the experimental results for the proposed methods and the stateof-the-art approaches. We can observe that LRSDP(w/o ρ)
obtains the very poor results, which reveals the significance of normalization using the logistic function. Moreover, the proposed LRSDP approach outperforms the two
recent social recommendation approaches: STE and SoRec.
This demonstrates that the graph Laplacian regularization
not only stands on a solid theoretical framework but also
obtains the promising results in practice. Looking into the
performance comparisons, we can also find that LRSDP(w/o
L) performs slightly better than its counterpart PMF method
using gradient descent optimization.
Finally, we empirically study the efficiency performance
of the proposed LRSDP method. Table 5 summarize the
computational time for factorizing the user-item rating matrix using STE and LRSDP. From these results, it can be
clearly observed that the proposed LRSDP approach is much
Performance Measure
Both the Root Mean Square Error (RMSE) and the Mean
Absolute Error (MAE) are employed as the performance
measure to evaluate the prediction quality of our proposed
approaches in comparison with other related recommendation methods. More specifically, the metrics RMSE is defined as follows:
1
(Mi,j − M̂i,j )2
RMSE =
p i,j
where Mi,j denotes the rating on the item j annotated by
the user i, and M̂i,j is the predicted rating for the user i on
the item j. p is denoted as the total number of ratings in the
testing dataset. Additionally, the metrics MAE is defined as
below:
1
|Mi,j − M̂i,j |
MAE =
p i,j
To make a consistent comparison with the previous methods,
we employ the inverse mapping function of g(x) in Eqn. 2
to transform the predicted rating value in our method back
into the original range.
5.3
Experiments on Epinions Dataset
The rank d is an important parameter for the matrix factorization problem. Instead of assigning it to an empirical value
162
Table 4: Performance Comparisons (A Smaller RMSE or MAE Value Means a Better Performance)
Training Data Metrics
PMF
SoRec
Trust
STE
LRSDP (w/o ρ ) LRSDP (w/o L) LRSDP
RMSE 1.1826 1.1530 1.2140 1.1346
1.4998
1.1502
1.1304
80%
MAE
0.8951 0.8638 0.9221 0.8594
1.1730
0.8830
0.8557
RMSE 1.1575 1.1333 1.1959 1.1109
1.5677
1.1292
1.1095
90%
MAE
0.8676 0.8442 0.9054 0.8377
1.2219
0.8593
0.8338
more efficient than the STE method (Ma, King, and Lyu
2009). Specifically, LRSDP only requires no more than 7%
of the training time for STE. This is because the LRSDP
approach employs an efficient quasi-Newton optimization
algorithm while the STE method involves with the timeconsuming step to directly fuse the social trust information
into the high dimensional user-item matrix. Additionally, we
can see that the computational time for the presented LRSDP
method increases along with the total number of ratings in
the user-item matrix.
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Table 5: Comparisons of time cost on Epinions dataset
STE (90%) LRSDP (80%) LRSDP (90%)
133min
7.5min
8.3min
6
Conclusions
It is clear that our novel low-rank semidefinite program approach to social recommendation is powerful and effective.
It offers several distinct advantages over the conventional
approaches. First, we introduce the graph Laplacian to effectively regularize the user-specific latent space and capture the underlying relationships among the different users.
Second, the presented social recommendation with the graph
Laplacian regularization problem is directly formulated into
the low-rank semidefinite programming, which can be efficiently solved by the quasi-Newton algorithm. Finally,
the mapping function for the normalization is carefully addressed in our formulation. Our approach has been tested
on the Epinions dataset with over half million ratings. The
encouraging experimental results show that our presented
method is both effective and promising.
In the future, we will investigate the relationship among
the items by taking into account of the category information.
Moreover, we will explore the recommendation problem in
the multimedia domain, in which the content information
in music and videos can be used to estimate the similarity
between the different items.
Acknowledgments
The work was fully supported by Program for New Century
Excellent Talents in University (NCET-09-0685) and Fundamental Research Funds for the Central Universities.
References
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